326 Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004
Particle Ratio Fluctuations and Isobaric Ensemble
for Chemical Freeze-out Scenario
Licinio L. S. Portugal
Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Cx.P. 68528, CEP 21945-970, Rio de Janeiro, RJ, Brazil
Received on 15 August, 2003.
In this work, we study the behavior of event-by-event distribution of particle ratios in Gran Canonical and Isobaric ensemble.We show that the experimental value of the width in theK/πratio distribution is much wider than that given by a unique grand canonical ensemble or Isobaric ensembe and maybe a mixture is required.
1
Introduction
The main purpose of the relativistic heavy ion program at CERN and RHIC is to study the properties of matter in hadronic interactions at very high energy density. Among many novel phenomena expected, the observation of the quark gluon plasma (QGP) predicted by the QCD is one of the main goals. For these studies, a crucial factor is to know whether or not the system reaches thermal equilibrium at some stage of the collision. In this respect, thermal mod-els are extensively used to analyze the final hadronic abun-dances [1]. In these models, it is assumed that there exists a stage where the chemical abundances of hadrons (includ-ing resonances) are suddenly frozen out, keep(includ-ing the exact memory of the last instant of the chemical equilibrium. Such stage is referred to as “chemical freeze-out” and the ob-served hadronic abundances are completely determined by the temperature T and baryonic chemical potentialµB of
the system at this point.
The above picture is equivalent to say that the hadronic gas at the chemical freeze-out is described by a grand canon-ical ensemble (GC). The fact is that the observed particle ratios are remarkably well reproduced adjusting just two pa-rameters,TandµB. Such fits have been done for4πNA49
data and for RHIC Au+Au data in a rapidity window of ra-pidity. Although the resulting fits are impressive[1], it is very difficult to imagine that final hadrons are in a global chemical equilibrium (rapidity distributions of particles, es-pecially those of hyperons, are far from constant). It may well be possible that to have a nice fit only for the chem-ical abundances does not imply that T and µB obtained
correspond to those for arealchemical equilibrium. They could be just some effective parameters to characterize cer-tain average of statistical ensembles corresponding to differ-ent physical conditions for chemical freeze-out.
In order to clarify the above mentioned aspect, we should go one step further than the mean values of chemical abundances, and study the behavior of less inclusive quan-tities such as event-by-event fluctuations and correlations of chemical abundances among different hadrons. In this
pa-per, we analyze the event-by-event fluctuations ofK/π ra-tio in the thermal model. The distribura-tion ofK/πratio is measured forN A49experiment. We show that the pure sta-tistical fluctuation given by a unique GC ensemble of reso-nance gas is much smaller than the experimental data. This imply that the observed particle abundances can not be re-produced in terms of one single GC ensemble, but we need surely a mixture of many statistical ensembles. To explain the observed width of the distribution ofK/π ratio[2], we have to fluctuate, for example, the chemical potentialµB.
For nucleus-nucleus collisions, the mixture of differentµB
is rather natural because of the baryons in the incident nuclei and fluctuating initial conditions[3].
When we consider the fluctuation of the ratio K/π, it is important to see if there exists any correlation amongK
andπ. For a GC ensemble of ideal hadronic gas, there ex-ists no correlation for the production ofK andπ. For an infinite system these assumptions really do not matter, but when one consider a domain of particle production with fi-nite volume (fireball) it is not clear at all whether a GC en-semble represents the best scenario. In particular for kaons, the strangeness conservation imposes some effects. Analy-sis based on the canonical ensemble has been done by Koch [4]. Another possible approach is the use of isobaric en-semble (ISO). This enen-semble describe a system with fixed pressure allowing volume fluctuations.
chemi-Licinio L. S. Portugal 327
cal freeze-out should be a local process, it seems natural that its mechanism is governed by the intensive parameters. Sta-tistically speaking, the observed chemical freeze-out state may correspond to the state of maximum entropy under a fixedpressure, and not under a fixed volume. It is worth-while to investigate if there exist differences between these two scenarios in the particle abundances, especially in the distribution ofK/πratio.
2
Grand Canonical Ensemble
Let us consider first the case of GC ensemble of hadronic resonances. For simplicity, we consider ideal Boltzmann gas of hadronic resonances, neglecting the quantum statis-tics and the effect of excluded volume. Except for pions, for temperatures we are interested, the Boltzmann gas ap-proximation is quite well. Furthermore, since we consider just the ratio of particle multiplicities, the effect of excluded volume is not very important. Under these approximations, the probabilityP(N1, ...Nh)of havingN1 particles of the
hadron 1,N2particles of the hadron2, and so on, tillh-th
hadron species is given as just the product of Poisson distri-butions for each hadronic species,
P(N1, ...Nh) = PGC(N1, ...Nh;T, µB) = (1)
=
h
Y
i=1 1
Ni!
[φiV]Nje−V φi,
whereirefers to the hadron species,V is the volume and
φi=
g
2π2m 3
i
K2(βmi)
βmi
eµiβ (2)
with β = 1/T and µi is the chemical potential of the
i−thhadron. It is given by
µi=BiµB+SiµS+Ti(3)µ3 (3)
withBi, Si andTi(3) are baryon number, strangeness and
3-rd component of isospin of the i − th hadron. The strangeness and isospin chemical potentialsµS andµ3 are
determined using the charge and strangeness neutrality. mi
andgiare the mass and statistical factor of the hadron, and
K2is the modified Bessel function.
The observed particle abundances are not directly given by the thermal multiplicities. We should take into account the feeding stream from the resonance decays. The multi-plicity of a stable (observable) hadron species, sayα, in a single collision event is then given by
Nobs α =
X
i
Ni→αNi, (4)
whereNi→α is the number ofαhadron produced by the
decay chain of the parent hadroni. If we consider the de-cay chain has no correlation with the thermal production mechanism, we may safely substitute Ni→α (integers) in
terms of their average values Fi→α (not necessarily
inte-gers). We constructed the table for allFi→α´sfollowing the
decay chain of thei−thhadron using the values given in the particle data table [5]. The distribution of the particle ratio,rαβis then calculated to be
P¡
rα/β¢=
* X
{Ni}
P(N1, ...Nh)δ
µ
rαβ−
¯
Nα
¯
Nβ
¶+
, (5)
where hOi denotes the average over collisional events of an observable O. In practice, we introduce a finite bin size ∆ for the variable rα/β and calculate the
probabil-ity of finding an event within a given bin specified by
[rαβ−∆/2, rαβ+ ∆/2]. The average number of a−th
hadron is
hN¯αi=
X
i
hNiiFi→α (6)
where in the case of GC, we have
hNii=φiV.
In Fig. 1, we show the result of Monte Carlo evaluation ofP¡
rK/π
¢
for the Pb+Pb central collision.
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.1
1 10
T=165 MeV GC
µΒ=180.61 MeV µΒ=219.70 MeV µΒ=241.85 MeV µΒ=271.18 MeV µΒ=288.58 MeV Exp NA49
Figure 1. Grand canonical distributions ofK/πratio. Different curves correspond to different chemical potential indicated. Black circles are experimental data.
Thermal parameters{T, µB}are determined to give the best
χ2-fit to the particle ratios of stable hadrons as done in Ref.
[1]. The value ofχ2is around 20, which is comparable to
that of [1]. This means that the average particle ratios are very well reproduced. However, as we see, the calculated distribution is very much narrower than the experimental value. The data [2] refers to a narrow rapidity window, and the corrections from detector acceptance are not included we may not consider the value shown here as definitive but the difference between the value from a single GC ensemble seems to be too large. It may be possible that the contamina-tion of kaons from the neighboring rapidity windows causes such a broadening of the distribution [4]. Therefore the ob-served large width may be, is due to the mixture of many GC with different thermodynamical parameters, {T, µB}. For
Pb+Pb collisions at SPS, the baryonic chemical potentialµB
328 Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004
ratio changes with the chemical potential for fixed temper-ature. This results indicates that, the observed width of the
K/πratio distribution requires a mixture of GC ensemble of appreciably different parameters.
3
Isobaric Ensemble
As mentioned in Introduction, it maybe possible that the chemical freeze-out mechanism is governed by a pressure value than a fixed volume as constraint. The isobaric parti-tion funcparti-tion is given as
ZP(β, µ, P) =
Z ∞
0
dV e−βP VZGC(β, µ, V) (7)
whereZGC(β, µ, V)is the partition function of the GC
en-semble. For a Boltzmann ideal gas, we can perform the in-tegral easily to get
ZP(β, µ, P) =
1
βP−P
iφie−P v0β
, (8)
where we have introduced the excluded volumev0 here as
in [6].
It can be shown that the probability function correspond-ing to Eq.(1) becomes
PISO(N1...Nh;T, P) = (1−
X
i
ηi)N!
Y
j
ηjNj
Nj!
, (9)
where
ηi=
φie−βP v0
βP . (10)
The single particle multiplicity distribution is then given by
PISO(Nj) =
X
N16=Nj
... X
Nh6=Nj
PISO(N1...Nh) =
= (1−ζj)ζjNj (11)
where:
ζj= ηj
1−P
i6=j
ηi
.
which is a geometric distribution.
Different from the GC case, the convolution of geomet-ric distributions is not a geometgeomet-ric distribution, but so-called negative binomial distribution. Suppose that the final state of nuclear collision is formed ofk independent ’fireballs’, described by the same isobaric ensemble specifiedTandP. LetNi(q) be the number ofi−thhadron contained in the
q−th”fireball”. Then the total number ofi−thhadron is
X
q
Ni(q)=Ni. (12)
In this case, the final probability distribution of having a set of hadrons, {N1, ..., Nh}, is given as the convolution
of Eq.(9),
PISO(k)(N1, ..., Nα) =
X · · ·X
nP
qN
(q)
i =Ni
o
" k
Y
q=1
P(N1q, ..., N
q h).
#
,
(13) where the summations are taken over all(N1q, ..., N
q h)
sat-isfying the constraints Eq.(12). Correspondingly, the single particle distribution Eq.(11) is modified as[7]
PISO(k) (Nα) = (1−ζα)k ζαNα
(Nα+k−1)!
Nα!(k−1)!
. (14) The average multiplicity ofi−thhadron is then
hNii=k
ζi
1−P
iζi
. (15)
The particle ratio and its distribution are calculated by evaluating Eq.5) by the Monte Carlo method, generating
{Ni} which obey the probability distribution Eq.(13). For
this purpose, use of the conditional probabilities for the ISO ensemble is found to be efficient. The probability of havingNi particles fori−thhadron after having already
N1particles for the hadron 1,N2particles for the hadron 2,
and so on tillNi−1is
P(Ni||N1, N2, .., Ni−1) = (1−η¯i)Zi+1 (Zi+Ni)!
Zi!Ni!
¯
ηNi
i ,
(16) whereZi =Pij−=11Nj andη¯i=ηi/
³ 1−P
j>iηj
´
.From this, we can evaluate Eq.(5) very efficiently. In Fig.2, we show the calculated distribution ofK/πratio for the ISO en-semble for several clusters. The ISO enen-semble gives a larger width compared to the GC ensemble, especially for smaller numbers of clusters. This can be understood in terms of the positive correlation amongKandπmultiplicities in the ISO ensemble, as mentioned before. On the other hand, for large number of clusters, the convoluted distribution Eq. (13) tends to the GC ensemble. This is a natural consequence of the statistical central limit theorem. However, different from the GC ensemble, for a given set ofT an P, the value of chemical potential changes to keep the baryon number con-servation so that the average values ofK/πvalue changes for different number of clusters in the system.
4
Discussion and Perspectives
Licinio L. S. Portugal 329
0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
1E-4 1E-3 0.01
ISO T=165 MeV P=78.40 MeV
K=1 K=3 K=5 K=10
Figure 2. Isobaric distributions ofK/πratio. Different curves cor-respond to different number of clusters as indicated.
the other hand, the isobaric ensemble seems to be more ade-quate for the description of the chemical freeze-out scenario. It already furnishes a wider distribution ofK/πratio due to the correlations in particle multiplicities. On the other hand, if the final state of the system is composed of many clus-ters, convolution of the ISO distributions from these clusters
tends to that of the GC distribution. However, in this case, the fluctuation of number of clusters with fixed(T, P)can also contribute to the width of theK/πdistribution. A fur-ther investigation in this line is in progress.
Acknowledgments
This work has been developed in collaboration with Drs. A.G. Grunfeld, C.E. Aguiar and T. Kodama and is supported by CNPq, CAPES, FAPERJ and FAPESP
References
[1] See for example, P. Braun-Munzingeret al., nucl-th/0304013.
[2] S.V. Afanasiev,et. al., Phys. Rev. Lett.86, 1965 (2001).
[3] C.E. Aguiar, Y. Hama, T. Kodama, T. Osada, AIP Conf.Proc.631:686-694,2003
[4] A. Majumder and V. Koch, nucl-th/0305047.
[5] Review of Particle Physics, Phys. Rev. D66(2002).
[6] G. D. Yen, M. Gorenstein, W. Greiner and S. N. Yang, Phys. Rev. C56, 2210 (1997).
